How to Solve Mathematical Problems [Paperback]

Wayne A. Wickelgren (Author)

Book Description

Publication Date: January 30, 1995

Seven problem-solving techniques include inference, classification of action sequences, subgoals, contradiction, working backward, relations between problems, and mathematical representation. Also, problems from mathematics, science, and engineering with complete solutions.

Product Details

• Paperback: 272 pages
• Publisher: Dover Publications; First Edition edition (January 30, 1995)
• Language: English
• ISBN-10: 0486284336
• ISBN-13: 978-0486284330
• Product Dimensions: 8.5 x 5.4 x 0.6 inches
• Shipping Weight: 11.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (4 customer reviews)
• Amazon Best Sellers Rank: #707,058 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

10 of 10 people found the following review helpful:

5.0 out of 5 stars Great introductory book to Problem Solving, June 23, 2001

By A Customer

This review is from: How to Solve Mathematical Problems (Paperback)

Buy this and Paul Zeitz's "Art and Craft of Problem Solving" and you'll be on your way to being addicted to challenging Mathematical Problems. It took a few readings for me to truly appreciate this book since it is an introductory book to mathematical problems. Many of the ideas presented are second nature to an experienced solver, but the lucid, clear presentation would be an excellent learning experience for the beginner. Topics discussed include Induction, definition of problems to find and problems to prove, avoiding "running around in circles," hill climbing, and a few others. I Very much recommend this book.

16 of 20 people found the following review helpful:

3.0 out of 5 stars Good on Details, but Poorly Organized, December 29, 2001

By 

Martin P. Cohen "Seeker" (Langhorne, PA USA) - See all my reviews
(REAL NAME)   

This review is from: How to Solve Mathematical Problems (Paperback)

This book analyzes several problems, mostly in recreational mathematics, in fine detail. One feature worthy of emulation is that the book will present a problem, ask the reader to try to solve it, provide some analysis, ask the reader to again try to solve it and repeat this procedure for several iterations.

That said, the global organization of the book leaves much to be desired. It opens by showing several example problems similar to others that are solved in the book. However, some of these initial examples are not later solved. There was one problem in particular, a chess problem, that I spent some time on unsuccessfully trying to solve, whose answer would have been appreciated.

Wicklegren uses an artificial intelligence paradigm in the organization of the book. While AI techniques are useful for computers, there are better pattern matching techniques more suitable for use by humans. Hill climbing, for example, which is given as a basic technique, is good for use by a computer when no better method can be found. However, it is not well suited for hand calculation. Wicklegren tries to cover over this by saying that any technique that solves a problem by simplifying it, is an example of hill climbing even if there is no associated metric. There are several other places where the book tries unsuccessfully to shoehorn solution strategies into the few general techniques around which the book is organized. For example, the use of restraints is given as an example of proof by contradiction. Recursion and induction are lumped into a chapter on the use of subgoals.

By training the author is a psychologist who apparently took a lot of courses in math. This is a good background for studying problem solving and someday someone with a similar background may write a worthwhile book on the subject. In this book, it is painfully obvious that the author's math skills are a bit rusty and that the manuscript was not reviewed by anyone whose math skills are more current. For example, one of the solutions given is incorrect. This is due to a mental lapse where playing cards with numbers 2 to 8 are treated as if there are 8 cards instead of 7. In the last chapter, which departs from recreational mathematics, there are a few mathematical examples whose solutions are overly complicated. The most flagrant example of this is a problem that asks to determine the height of a triangle given the length b of the base and the two base angles of A and B. The straightforward solution to this is to consider the two segments that the altitude divides the base into to get the equation b = h*cot A + h*cot B. Instead, the book uses Heron's formula for the area of a triangle. The book presents an algebraic proof of Pascal's Identity without labeling it as such and without reference to the much more intuitive combinatorial proof.

Overall, I would barely recommend this book because of the way that the details are presented, but I would advise the reader to ignore the surrounding contextual presentation.

3 of 3 people found the following review helpful:

4.0 out of 5 stars neat strategies, October 27, 2005

By 

W Boudville (Terra, Sol 3) - See all my reviews
(VINE VOICE)    (TOP 1000 REVIEWER)    (HALL OF FAME REVIEWER)    (REAL NAME)   

This review is from: How to Solve Mathematical Problems (Paperback)

At first sight, this might seem like a book to solve math party puzzles. Indeed it can be used for that. But it also lets you understand strategies that may be applied in any context that a maths problem can arise. For instance, there is the idea of systematic trial and error. And a discussion of how to analyse what methods you have been using to tackle a problem. Very useful, because it might suggest other methods, simply by having you explicitly recap your previous methods.

As the book says, think about what you did, rather than about the problem. As an inventor, this is just what I do. Plus, the book recommends "incubation", where if you are making no headway, you put aside the problem for some time. Hours or days. Then, just maybe, your subconscious might operate on it during this downtime. So that when you later explicitly return to the problem, a solution emerges. This method is sometimes commonly known as "sleep on it".

Wickelgren cautions that this idea of your subconscious working on a problem might be bunk. He gives plausible alternatives. Like simply being less fatigued, physically and mentally, when you return to the problem after a hiatus.